I'm getting into a tangle over absolutely elementary definitions in Category theory.
Consider an arbitrary category with some number $n$ of objects (say $n > 2$). Pick two objects in the Category, say $A$ and $B$. Does there (by definition) exist a morphism between $A$ and $B$?
I'd always thought the answer was yes (until I actually started studying some Category theory in earnest - today). I now think the answer is no.